Read e-book online Applications of Lie Groups to Differential Equations PDF

By Peter J. Olver

ISBN-10: 1468402749

ISBN-13: 9781468402742

This e-book is dedicated to explaining quite a lot of purposes of con­tinuous symmetry teams to bodily very important platforms of differential equations. Emphasis is put on major purposes of group-theoretic tools, prepared in order that the utilized reader can quite simply study the fundamental computational suggestions required for actual actual difficulties. the 1st bankruptcy collects jointly (but doesn't turn out) these features of Lie workforce thought that are of value to differential equations. functions lined within the physique of the e-book comprise calculation of symmetry teams of differential equations, integration of normal differential equations, together with targeted recommendations for Euler-Lagrange equations or Hamiltonian platforms, differential invariants and development of equations with pre­scribed symmetry teams, group-invariant suggestions of partial differential equations, dimensional research, and the connections among conservation legislation and symmetry teams. Generalizations of the fundamental symmetry crew idea, and functions to conservation legislation, integrability stipulations, thoroughly integrable structures and soliton equations, and bi-Hamiltonian structures are lined intimately. The exposition is fairly self-contained, and supplemented through quite a few examples of direct actual significance, selected from classical mechanics, fluid mechanics, elasticity and different utilized components.

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Additional info for Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, Volume 107)

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Introduction to Lie Groups where v2 (f) = v( v(f», v3 (f) = v( v2 (f», etc. If we assume convergence of the entire Taylor series in e, then we obtain the Lie series 00 f(exp(ev)x) ek = k~O k! 18) for the action of the flow on f. The same result holds for vector-valued functions F: M ~ IIl n, F(x) = (FJ(x), ... , Fn(x», where we let v act component-wise on F: v(F) = (v(F J ), ••• , v(Fn». In particular, if we let F be the coordinate functions x, we obtain (again under assumptions of convergence) a Lie series for the flow itself, given by e2 00 ek exp(ev)x=x+eg(x)+-v(~")(x)+"'= L -k vk(x), 2 k=O !

Introduction to Lie Groups in the y-coordinates. 5) holds if and only if d dde I/I(cp(e*)) = dO 1/1(4)(0*)), which proves the claim. 4) behaves under the given change of coordinates y = I/I(x). The collection of all tangent vectors to all possible curves passing through a given point x in M is called the tangent space to M at x, and is denoted by TMlx. If M is an m-dimensional manifold, then TMlx is an mdimensional vector space, with {a/ axl, ... ,a/ axm} providing a basis for TMlx in the given local coordinates.

If H is a closed subgroup of G, then H is a regular submanifold of G and hence a Lie group in its own right. Conversely, any regular Lie subgroup of G is a closed subgroup. Note that we need only check that H is a subgroup of G and is closed as a subset of G in order to conclude that H is a regular Lie subgroup. This circumvents the problem of actually proving that H is a submanifold. In particular, if H is a subgroup defined by the vanishing of a number of (continuous) real-valued functions H = {g: Fi(g) = 0, i = 1, ...

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Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, Volume 107) by Peter J. Olver

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